1

State-Space: Controllability & Observability

&Information Theory

& More!

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh (with material from Dr. Paul Pounds)

Lecture 24

elec3004@itee.uq.edu.auhttp://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

May 29, 2013

State-Space:Made Clear

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh (with material from Dr. Paul Pounds)

Lecture 24

elec3004@itee.uq.edu.auhttp://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

May 29, 2013

Made Clear

2

Today in Linear Systems…

ELEC 3004: Systems 29 May 2013 - 3

Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control17-MayApplications in Industry

1222-MayState-Space24-MayControllability & Observability

13 29-MayState-Space: Made Clear31-MaySummary and Course Review

Today:• Review State-Space

• Quick introduction to Information Theory and Communications

Friday:• Everything

(Literally!)

ELEC 3004: Systems 29 May 2013 - 4

Goals for the Week

3

• Practice Final Posted:– You’re welcome – While it’s final #3 (real, supp, & prac):

• It shares no questions with either the real or the supp.

– Exam Review Session: June 7,

It is harder than the real thing!(∵ its primarily to practice the material not the timing)

• Problem Set 2 Grading:– May 31 (& you get your grades during the semester)– June 7 (More time, better feedback??)

ELEC 3004: Systems 29 May 2013 - 5

Announcements:

!

State-space control design¿¿¿Que pasa????

• Design for discrete state-space systems is just like the continuous case.– Apply linear state-variable feedback:

such that detwhere is the desired control characteristic equation

Predictably, this requires the system controllability matrix ⋯ to be full-rank.

29 May 2013 -ELEC 3004: Systems 6

4

• Recall:

• For Linear Systems:

• For LTI:

ELEC 3004: Systems 29 May 2013 - 7

Solving State Space…

• In the conventional, frequency-domain approach the differential equations are converted to transfer functions as soon as possible– The dynamics of a system comprising several subsystems is

obtained by combining the transfer functions!

• With the state-space methods, on the other hand, the description of the system dynamics in the form of differential equations is retained throughout the analysis and design.

ELEC 3004: Systems 29 May 2013 - 8

Solving State Space

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• Describes how the state x(t) of the system at some time t evolves into (or from) the state x(τ) at some other time T.

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State-transition matrix Φ(t)

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Solving State Space…

6

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Solving State Space

ELEC 3004: Systems 29 May 2013 - 12

Solving State Space

7

ELEC 3004: Systems 29 May 2013 - 13

Solving State Space

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Solving State Space

8

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Solving State Space

• Difference equations in state-space form:

• Where:– u[n], y[n]: input & output (scalars)– x[n]: state vector

ELEC 3004: Systems 29 May 2013 - 16

Digital State Space:Recall from the Last Episode …

9

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Digital Control Law Design

•Yes

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Can you use this for more than Control?

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The Approach:• Formulate the goal of control as an optimization (e.g. minimal impulse response,

minimal effort, ...).• You’ve already seen some examples of optimization-based design:

– Used least-squares to obtain an FIR system which matched (in the least-squares sense) the desired frequency response.

– Poles/zeros lecture: Butterworth filter

ELEC 3004: Systems 29 May 2013 - 19

Frequency Response in State Space

ELEC 3004: Systems 29 May 2013 - 20

Discrete Time Butterworth Filters

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ELEC 3004: Systems 29 May 2013 - 22

12

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ELEC 3004: Systems 29 May 2013 - 24

13

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ELEC 3004: Systems 29 May 2013 - 26

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• Constrained Least-Squares …

ELEC 3004: Systems 29 May 2013 - 27

How?

On the transmission of information over a noisy channel:• An information source that produces a message• A transmitter that operates on the message to create a signal which can be sent

through a channel• A channel, which is the medium over which the signal, carrying the information

that composes the message, is sent• A receiver, which transforms the signal back into the message intended for

delivery• A destination, which can be a person or a machine, for whom or which the

message is intended

ELEC 3004: Systems 29 May 2013 - 28

Shannon Information Theory

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• AKA: The last lecture!

ELEC 3004: Systems 29 May 2013 - 29

Next Time in Linear Systems ….Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems

15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems

22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response

12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals

26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems

3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control

17-MayApplications in Industry

1222-MayState-Space

24-MayControllability & Observability

1329-MayState-Space: Made Clear

31-MaySummary and Course Review